Chapter 4.5, Problem 12E

To determine

To calculate: The solution of the exponential , rounded to four decimal places , $4\left(1+{10}^{5x}\right)=9$

Answer to Problem 12E

The four decimal places rounded value of exponential expression,$x=0.0194$

Explanation of Solution

Given information:

The given expression is as, $4\left(1+{10}^{5x}\right)=9$

Formula used:

For the exponential expression take log on both sides.

These are the steps to do this ,

  ${\log }_{10}\left[{10}^x\right]=x,\log e^x=x\log e$

Step 1 take log both side.

Step 2. Then to find the value of x one side take all the values of x and another side all the values of log and constants .

Step 3. Then with the help of log table put the value of log .

Step 4. Take the four decimal value of x from there .

Calculation :

Consider the expression, $4\left(1+{10}^{5x}\right)=9$

Take logarithms both sides.

  $\begin{array}{l}4\left(1+{10}^{5x}\right)=9\\ 1+{10}^{5x}=\frac{9}{4}\\ {10}^{5x}=\frac{9}{4}-1\\ {10}^{5x}=\frac{5}{4}\end{array}$

Apply the identity: ${\log }_{10}\left[{10}^x\right]=x$

  $\begin{array}{c}{\log }_{10}\left[{10}^{5x}\right]={\log }_{10}\left[\frac{5}{4}\right]\\ 5x={\log }_{10}\left[\frac{5}{4}\right]\\ x=\frac{{\log }_{10}\left[\frac{5}{4}\right]}{5}\end{array}$

Apply the value of ${\log }_{10}\left[\frac{5}{4}\right]$ :

Rewrite the expression: $\begin{array}{l}x=\frac{{\log }_{10}\left[\frac{5}{4}\right]}{5}\\ x=0.0194\end{array}$

The above expression can’t be simplified further.

Thus the simplified form of the expression is $x=0.0194$ .